A general framework for the optimal approximation of circular arcs by parametric polynomial curves

TL;DR

This paper introduces a comprehensive framework for optimally approximating circular arcs with parametric polynomial curves, utilizing constrained uniform approximation and nonlinear equations to achieve the best radial distance approximations.

Contribution

It presents a novel general approach for geometric approximation of circular arcs, including explicit solutions for low-degree cases and a conjecture on optimality.

Findings

Explicit solutions for low-degree approximations are derived.

The proposed method achieves the best known radial distance approximations.

Numerical examples support the theoretical framework.

Abstract

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.